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Differential Equations/First Partial Derivatives

Question
I am really struggling with these types... thank you in advance for your help!

Find all first partial derivatives of:

f(x,y,z)=arctan(y/xz)

f(x,y)=e^(xy) cos(x)

The derivative of f(t) = arctan(t) is (df/dt) = is 1/(1 + t^2),
and this will be used in finding each of t he partial derivatives at the bottom.

Now for y/xz, I will assume that is really y/(xz).
Math rules says to do division and mulitplication from left to right at the same time,
so y/xz turns out to be (y/x)z, which is really yz/x, and that doesn't look right.

I will let t = y/(xz).
From this, dt/dx = -y/(x²z), dt/dy = 1/(xz), dt/dz = -y/(xz²)

This means that the partial derivatives are ...
with respect to x, (df/dt)(dt/dx);
with respect to y, (df/dt)(dt/dy); and
with respect to z, (df/dt)(dt/dz).

Differential Equations

Volunteer

Scott A Wilson

Expertise

I am capable of solving most ordinary differential equations and a few not so ordinary, but those are few and far between.

Experience

I have taken Differential Equations at OSU. Occasionally I have assisted people with questions on the subject.

Publications
I wrote a thesis on the study of shock waves and rarefaction fans. That occurs in nature when mathematics fails to apply. See, mathematics says that when shock waves appear, there should be two solutions whereas nature knows there is only one way to be in that area. When rarefaction fans occur, the mathematics says there should be no solution, but despite this, nature still moves ahead with its own plan.

Education/Credentials
I received an MS degree at OSU in Mathemematics and a BS degree at OSU in Mathematical Science.

Awards and Honors
I graduated from OSU with a Bachelor of Science degree in mathematics with honors. Two years later, I graduated from OSU with a Master of Science degree in mathematics with honors. Between these two degrees, I have more hours in graduate courses that required to get them.

Past/Present Clients
I answered many questions at OSU, down in south seattle at a college, at a church in Corvallis, OR, as Safeway in Washington, and many other areas. I have answered over 8,500 right here, but only a little over 190 of them have been in differential equations.